Abstract

We study the Γ\Gamma-convergence as ε→0+\varepsilon \to 0^+ of the family of degenerate functionals Qε(u)=ε∫Ω⟨ADu,Du⟩ dx+1ε∫ΩW(u) dxQ_\varepsilon(u) = \varepsilon \int_\Omega \langle ADu, Du\rangle \, dx + \frac{1}{\varepsilon} \int_\Omega W(u) \, dx, where A(x) is a symmetric, non negative n×nn\times n matrix on Ω\Omega (i.e. ⟨A(x)ξ,ξ⟩≥0\langle A(x)\xi, \xi\rangle \ge 0 for all x∈Ωx \in \Omega and ξ∈Rn\xi \in \mathbb{R}^n) with regular entries and W:R→[0,+∞)W : \mathbb{R} \to [0, +\infty) is a double well potential having two isolated minimum points. Moreover, under suitable assumptions on the matrix A, we obtain a minimal interface criterion for the Γ\Gamma-limit functional exploiting some tools of analysis in Carnot-Caratheodory spaces. We extend some previous results obtained for the non degenerate perturbations QεQ_\varepsilon in the classical gradient theory of phase transitions.

Author information

Contact details are reproduced from the original publication and may be historical.

Francesco Serra Cassano

Dip. di Matematica, Università di Trento, Via Sommarive 14, 38050 Povo, Italy

cassano@science.unitn.it

Keywords

  • Phase transitions
  • Γ-convergence
  • Carnot-Carathéodory spaces
  • minimal interface criterion

Mathematics Subject Classification

49J45, 49Q05, 49Q20

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Suggested citation

R. Monti, F. Serra Cassano. “Degenerate Perturbations of a Two-Phase Transition Model.” Journal of Convex Analysis 10 (2003), No. 1, 1–34. https://doi.org/10.68381/jca1001

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press.